The tropical symplectic Grassmannian

TL;DR

This paper initiates the study of the tropicalization of the symplectic Grassmannian, exploring its properties, relations, and examples, and establishing foundational results and future directions in tropical geometry.

Contribution

It formulates tropical analogues of symplectic Grassmannian characterizations, determines when relations form a tropical basis, and analyzes properties of conormal fans of matroids.

Findings

Plücker and symplectic relations form a tropical basis if and only if rank ≤ 2

Several features of the symplectic Grassmannian do not survive tropicalization

Characterizations hold for conormal fans of matroids under certain conditions

Abstract

We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces that are isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Pl\"ucker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when do conormal fans of matroids satisfy these characterizations, as well as doing the same for a non-constant coefficient generalization. Finally, we propose several directions to extend the study of the tropical symplectic Grassmannian.